MathLabs

Problem 5

Let ABCDEF ABCDEF be a convex hexagon such that AB∥DE AB\parallel DE , BC∥EF BC\parallel EF , and CD∥FA CD\parallel FA . Let RA,RC,RE R_A,R_C,R_E be the circumradii of triangles FAB FAB , BCD BCD , DEF DEF , respectively, and let p p be the perimeter of the hexagon. Prove that RA+RC+RE≥p/2 R_A+R_C+R_E\ge p/2.
Step 4 of 4: Finish with AM-GM
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In plain words

Finish with AM-GM

4(RA+RC+RE)≥2p4(R_A+R_C+R_E)\ge2p
Detailed analysis

Each coefficient of a side in the sum has the form x+1/x x+1/x for a positive x x , hence is at least 22. The right side is therefore at least 2(AB+BC+CD+DE+EF+FA)=2p2(AB+BC+CD+DE+EF+FA)=2p . The left side is 4(RA+RC+RE)4(R_A+R_C+R_E), so RA+RC+RE≥p/2 R_A+R_C+R_E\ge p/2.